Geometry of adaptive control: optimization and geodesics

Abstract

Two incompatible topologies appear in the study of adaptive systems: the graph topology in control design, and the coefficient topology in system identification. Their incompatibility is manifest in the stabilization problem of adaptive control. We argue that this problem can be approached by changing the geometry of the sets of control systems under consideration: estimating n p parameters in an n p -dimensional manifold whose points all correspond to stabilizable systems. One way to construct the manifold is using the properties of the algebraic Riccati equation. Parameter estimation can be approached as an optimal control problem akin to the deterministic Kalman filter, leading to algorithms that can be used in conjunction with standard observers and controllers to construct stable adaptive systems. TOPOLOGIES IN ADAPTIVE CONTROLTwo topologies appear in the study of adaptive systems. Relevant for feedback control is the graph topology, induced by both the gap and graph metrics; it is the coarsest topology on sets of linear systems for which feedback stability is a robust property [1,2]. System identification, on the other hand, makes implicit use of the topology induced by the metric in which the distance between two systems is given by the Euclidean distance between the coefficients of their transfer functions. Even on sets of linear systems with dimension not exceeding a given n; on which both are defined, these topologies are not compatible: one is neither finer nor coarser than the other,

Control Lyapunov Function Design by Cancelling Input Singularity

“…For example, backstepping is applied to extend a small attraction region by feedback linearization, but the global stability is not guaranteed because the singularity of the resulting control system still remains [6]. Some methods can achieve the global stability by avoiding the singularity, for instance, variable structure control [8,11], sliding mode control [1,3], adaptive control [2], and fuzzy control [5]. The basic idea of these methods is to change the control input or the dynamics of the system for the trajectory of the control system to avoid the singularity when the trajectory approaches the singular manifold.…”

“…ε , 2 ε , 3 ε are design parameters [6]. The control input u in (13) includes a rational function, thus u is not defined on the singular manifold 1 2…”

Kalman Filter on Power Electronics and Power Systems Applications

“…Then several different applications progressively emerged in economics, image processing and biomedical instrumentation, to name a few. More recently, the KF found applications as part of more complex systems, as in an adaptive control system -see for example (Sastry & Bodson, 1989), where it is shown that the RLS (recursive least mean square) algorithm is a particular case of the KF -and attempts to find a nonlinear KF have been taking place, as in (Wong & Yau, 1999) and (Colón & Pait, 2004). This chapter presents a concise survey of applications of KF to power systems and power electronics, giving emphasis on the topic of signal's fundamental component identification, which has a key role in most of them.…”